In
mathematics, an associative algebra ''A'' is an
algebraic structure with compatible operations of addition, multiplication (assumed to be
associative), and a
scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
by elements in some
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''K''. The addition and multiplication operations together give ''A'' the structure of a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
; the addition and scalar multiplication operations together give ''A'' the structure of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over ''K''. In this article we will also use the term
''K''-algebra to mean an associative algebra over the field ''K''. A standard first example of a ''K''-algebra is a ring of
square matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
over a field ''K'', with the usual
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
.
A commutative algebra is an associative algebra that has a
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
multiplication, or, equivalently, an associative algebra that is also a
commutative ring.
In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures
non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital.
Many authors consider the more general concept of an associative algebra over a
commutative ring ''R'', instead of a field: An ''R''-algebra is an
''R''-module with an associative ''R''-bilinear binary operation, which also contains a multiplicative identity. For examples of this concept, if ''S'' is any ring with
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
''C'', then ''S'' is an associative ''C''-algebra.
Definition
Let ''R'' be a
commutative ring (so ''R'' could be a field). An associative ''R''-algebra (or more simply, an ''R''-algebra) is a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
that is also an
''R''-module in such a way that the two additions (the ring addition and the module addition) are the same operation, and
scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
satisfies
:
for all ''r'' in ''R'' and ''x'', ''y'' in the algebra. (This definition implies that the algebra is
unital, since rings are supposed to have a
multiplicative identity.)
Equivalently, an associative algebra ''A'' is a ring together with a
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preser ...
from ''R'' to the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
of ''A''. If ''f'' is such a homomorphism, the scalar multiplication is
(here the multiplication is the ring multiplication); if the scalar multiplication is given, the ring homomorphism is given by
(See also below).
Every ring is an associative
-algebra, where
denotes the ring of the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s.
A is an associative algebra that is also a
commutative ring.
As a monoid object in the category of modules
The definition is equivalent to saying that a unital associative ''R''-algebra is a
monoid object
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object ''M'' together with two morphisms
* ''μ'': ''M'' ⊗ ''M'' → ''M'' called ''multiplication'',
* ''η' ...
in
''R''-Mod (the
monoidal category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left a ...
of ''R''-modules). By definition, a ring is a monoid object in the
category of abelian groups; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the
category of modules.
Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in some other category that behaves like the category of modules. Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra ''A''. For example, the associativity can be expressed as follows. By the universal property of a
tensor product of modules In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor produc ...
, the multiplication (the ''R''-bilinear map) corresponds to a unique ''R''-linear map
:
.
The associativity then refers to the identity:
:
From ring homomorphisms
An associative algebra amounts to a
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preser ...
whose image lies in the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
. Indeed, starting with a ring ''A'' and a ring homomorphism
whose image lies in the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
of ''A'', we can make ''A'' an ''R''-algebra by defining
:
for all ''r'' ∈ ''R'' and ''x'' ∈ ''A''. If ''A'' is an ''R''-algebra, taking ''x'' = 1, the same formula in turn defines a ring homomorphism
whose image lies in the center.
If a ring is commutative then it equals its center, so that a commutative ''R''-algebra can be defined simply as a commutative ring ''A'' together with a commutative ring homomorphism
.
The ring homomorphism ''η'' appearing in the above is often called a
structure map. In the commutative case, one can consider the category whose objects are ring homomorphisms ''R'' → ''A''; i.e., commutative ''R''-algebras and whose morphisms are ring homomorphisms ''A'' → ''A'' that are under ''R''; i.e., ''R'' → ''A'' → ''A'' is ''R'' → ''A'' (i.e., the
coslice category of the category of commutative rings under ''R''.) The
prime spectrum functor Spec then determines an anti-equivalence of this category to the category of
affine schemes over Spec ''R''.
How to weaken the commutativity assumption is a subject matter of
noncommutative algebraic geometry and, more recently, of
derived algebraic geometry Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over \mathbb), simplicial commutati ...
. See also:
generic matrix ring.
Algebra homomorphisms
A
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
between two ''R''-algebras is an
''R''-linear ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preser ...
. Explicitly,
is an associative algebra homomorphism if
:
The class of all ''R''-algebras together with algebra homomorphisms between them form a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
, sometimes denoted ''R''-Alg.
The
subcategory of commutative ''R''-algebras can be characterized as the
coslice category ''R''/CRing where CRing is the
category of commutative rings
In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is ...
.
Examples
The most basic example is a ring itself; it is an algebra over its
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
or any subring lying in the center. In particular, any commutative ring is an algebra over any of its subrings. Other examples abound both from algebra and other fields of mathematics.
Algebra
*Any ring ''A'' can be considered as a Z-algebra. The unique ring homomorphism from Z to ''A'' is determined by the fact that it must send 1 to the identity in ''A''. Therefore, rings and Z-algebras are equivalent concepts, in the same way that
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s and Z-modules are equivalent.
*Any ring of
characteristic ''n'' is a (Z/''n''Z)-algebra in the same way.
*Given an ''R''-module ''M'', the
endomorphism ring
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a ...
of ''M'', denoted End
''R''(''M'') is an ''R''-algebra by defining (''r''·φ)(''x'') = ''r''·φ(''x'').
*Any ring of
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
with coefficients in a commutative ring ''R'' forms an ''R''-algebra under matrix addition and multiplication. This coincides with the previous example when ''M'' is a finitely-generated,
free ''R''-module.
**In particular, the square ''n''-by-''n''
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
with entries from the field ''K'' form an associative algebra over ''K''.
* The
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s form a 2-dimensional commutative algebra over the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s.
* The
quaternions form a 4-dimensional associative algebra over the reals (but not an algebra over the complex numbers, since the complex numbers are not in the center of the quaternions).
* The
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s with real coefficients form a commutative algebra over the reals.
* Every
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
''R''
1, ..., ''xn''">'x''1, ..., ''xn''is a commutative ''R''-algebra. In fact, this is the free commutative ''R''-algebra on the set .
* The
free ''R''-algebra on a set ''E'' is an algebra of "polynomials" with coefficients in ''R'' and noncommuting indeterminates taken from the set ''E''.
* The
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
of an ''R''-module is naturally an associative ''R''-algebra. The same is true for quotients such as the
exterior
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of th ...
and
symmetric algebra
In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
s. Categorically speaking, the
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
that maps an ''R''-module to its tensor algebra is
left adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
to the functor that sends an ''R''-algebra to its underlying ''R''-module (forgetting the multiplicative structure).
*The following ring is used in the theory of
λ-rings. Given a commutative ring ''A'', let
the set of formal power series with constant term 1. It is an abelian group with the group operation that is the multiplication of power series. It is then a ring with the multiplication, denoted by
, such that
determined by this condition and the ring axioms. The additive identity is 1 and the multiplicative identity is
. Then
has a canonical structure of a
-algebra given by the ring homomorphism
On the other hand, if ''A'' is a λ-ring, then there is a ring homomorphism
giving
a structure of an ''A''-algebra.
Representation theory
* The universal enveloping algebra of a Lie algebra is an associative algebra that can be used to study the given Lie algebra.
* If ''G'' is a group and ''R'' is a commutative ring, the set of all functions from ''G'' to ''R'' with finite support form an ''R''-algebra with the convolution as multiplication. It is called the
group algebra of ''G''. The construction is the starting point for the application to the study of (discrete) groups.
* If ''G'' is an
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Ma ...
(e.g., semisimple
complex Lie group), then the
coordinate ring
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal ...
of ''G'' is the
Hopf algebra ''A'' corresponding to ''G''. Many structures of ''G'' translate to those of ''A''.
* A
quiver algebra (or a path algebra) of a directed graph is the free associative algebra over a field generated by the paths in the graph.
Analysis
* Given any
Banach space ''X'', the
continuous linear operators ''A'' : ''X'' → ''X'' form an associative algebra (using composition of operators as multiplication); this is a
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
.
* Given any
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'', the continuous real- or complex-valued functions on ''X'' form a real or complex associative algebra; here the functions are added and multiplied pointwise.
* The set of
semimartingale
In probability theory, a real valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the ...
s defined on the
filtered probability space
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
(Ω, ''F'', (''F
t'')
''t'' ≥ 0, P) forms a ring under
stochastic integration.
* The
Weyl algebra
In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form
: f_m(X) \partial_X^m + f_(X) \partial_X^ + \cdots + f_1(X) \partial_X + f_0(X).
More prec ...
* An
Azumaya algebra In mathematics, an Azumaya algebra is a generalization of central simple algebras to ''R''-algebras where ''R'' need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where ''R'' is a commutative local rin ...
Geometry and combinatorics
* The
Clifford algebras, which are useful in
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
.
*
Incidence algebra In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set
and commutative ring with unity. Subalgebras called reduced incidence algebras give a natural constructi ...
s of
locally finite partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
s are associative algebras considered in
combinatorics.
* The
partition algebra and its subalgebras, including the
Brauer algebra and the
Temperley-Lieb algebra.
Constructions
;Subalgebras: A subalgebra of an ''R''-algebra ''A'' is a subset of ''A'' which is both a
subring and a
submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
of ''A''. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of ''A''.
;Quotient algebras: Let ''A'' be an ''R''-algebra. Any ring-theoretic
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
''I'' in ''A'' is automatically an ''R''-module since ''r'' · ''x'' = (''r''1
''A'')''x''. This gives the
quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
''A'' / ''I'' the structure of an ''R''-module and, in fact, an ''R''-algebra. It follows that any ring homomorphic image of ''A'' is also an ''R''-algebra.
;Direct products: The direct product of a family of ''R''-algebras is the ring-theoretic direct product. This becomes an ''R''-algebra with the obvious scalar multiplication.
;Free products: One can form a
free product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and is ...
of ''R''-algebras in a manner similar to the free product of groups. The free product is the
coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduc ...
in the category of ''R''-algebras.
;Tensor products: The tensor product of two ''R''-algebras is also an ''R''-algebra in a natural way. See
tensor product of algebras
In mathematics, the tensor product of two algebras over a commutative ring ''R'' is also an ''R''-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of such products is to describe the prod ...
for more details. Given a commutative ring ''R'' and any ring ''A'' the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
''R'' ⊗
Z ''A'' can be given the structure of an ''R''-algebra by defining ''r'' · (''s'' ⊗ ''a'') = (''rs'' ⊗ ''a''). The functor which sends ''A'' to ''R'' ⊗
Z ''A'' is
left adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
to the functor which sends an ''R''-algebra to its underlying ring (forgetting the module structure). See also:
Change of rings.
Separable algebra
Let ''A'' be an algebra over a commutative ring ''R''. Then the algebra ''A'' is a right module over
with the action
. Then, by definition, ''A'' is said to
separable if the multiplication map
splits as an
-linear map, where
is an
-module by
. Equivalently,
is separable if it is a
projective module over
; thus, the
-projective dimension of ''A'', sometimes called the bidimension of ''A'', measures the failure of separability.
Finite-dimensional algebra
Let ''A'' be a finite-dimensional algebra over a field ''k''. Then ''A'' is an
Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are ...
.
Commutative case
As ''A'' is Artinian, if it is commutative, then it is a finite product of Artinian local rings whose residue fields are algebras over the base field ''k''. Now, a reduced Artinian local ring is a field and thus the following are equivalent
#
is separable.
#
is reduced, where
is some algebraic closure of ''k''.
#
for some ''n''.
#
is the number of
-algebra homomorphisms
.
Noncommutative case
Since a
simple Artinian ring is a (full) matrix ring over a division ring, if ''A'' is a simple algebra, then ''A'' is a (full) matrix algebra over a division algebra ''D'' over ''k''; i.e.,
. More generally, if ''A'' is a semisimple algebra, then it is a finite product of matrix algebras (over various division ''k''-algebras), the fact known as the
Artin–Wedderburn theorem.
The fact that ''A'' is Artinian simplifies the notion of a Jacobson radical; for an Artinian ring, the Jacobson radical of ''A'' is the intersection of all (two-sided) maximal ideals (in contrast, in general, a Jacobson radical is the intersection of all left maximal ideals or the intersection of all right maximal ideals.)
The Wedderburn principal theorem states: for a finite-dimensional algebra ''A'' with a nilpotent ideal ''I'', if the projective dimension of
as an
-module is at most one, then the natural surjection
splits; i.e.,
contains a subalgebra
such that
is an isomorphism. Taking ''I'' to be the Jacobson radical, the theorem says in particular that the Jacobson radical is complemented by a semisimple algebra. The theorem is an analog of
Levi's theorem for
Lie algebras.
Lattices and orders
Let ''R'' be a Noetherian integral domain with field of fractions ''K'' (for example, they can be
). A ''
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
'' ''L'' in a finite-dimensional ''K''-vector space ''V'' is a finitely generated ''R''-submodule of ''V'' that spans ''V''; in other words,
.
Let
be a finite-dimensional ''K''-algebra. An ''
order'' in
is an ''R''-subalgebra that is a lattice. In general, there are a lot fewer orders than lattices; e.g.,
is a lattice in
but not an order (since it is not an algebra).
A ''maximal order'' is an order that is maximal among all the orders.
Related concepts
Coalgebras
An associative algebra over ''K'' is given by a ''K''-vector space ''A'' endowed with a bilinear map ''A'' × ''A'' → ''A'' having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism ''K'' → ''A'' identifying the scalar multiples of the multiplicative identity. If the bilinear map ''A'' × ''A'' → ''A'' is reinterpreted as a linear map (i. e.,
morphism in the category of ''K''-vector spaces) ''A'' ⊗ ''A'' → ''A'' (by the
universal property of the tensor product), then we can view an associative algebra over ''K'' as a ''K''-vector space ''A'' endowed with two morphisms (one of the form ''A'' ⊗ ''A'' → ''A'' and one of the form ''K'' → ''A'') satisfying certain conditions that boil down to the algebra axioms. These two morphisms can be dualized using
categorial duality by reversing all arrows in the
commutative diagrams that describe the algebra
axioms; this defines the structure of a
coalgebra In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams ...
.
There is also an abstract notion of
''F''-coalgebra, where ''F'' is a
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
. This is vaguely related to the notion of coalgebra discussed above.
Representations
A
representation of an algebra ''A'' is an algebra homomorphism ''ρ'' : ''A'' → End(''V'') from ''A'' to the endomorphism algebra of some vector space (or module) ''V''. The property of ''ρ'' being an algebra homomorphism means that ''ρ'' preserves the multiplicative operation (that is, ''ρ''(''xy'') = ''ρ''(''x'')''ρ''(''y'') for all ''x'' and ''y'' in ''A''), and that ''ρ'' sends the unit of ''A'' to the unit of End(''V'') (that is, to the identity endomorphism of ''V'').
If ''A'' and ''B'' are two algebras, and ''ρ'' : ''A'' → End(''V'') and ''τ'' : ''B'' → End(''W'') are two representations, then there is a (canonical) representation ''A''
''B'' → End(''V''
''W'') of the tensor product algebra ''A
B'' on the vector space ''V
W''. However, there is no natural way of defining a
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of two representations of a single associative algebra in such a way that the result is still a representation of that same algebra (not of its tensor product with itself), without somehow imposing additional conditions. Here, by ''
tensor product of representations In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product. This construction, together with the Clebsch–Gordan procedure, can be ...
'', the usual meaning is intended: the result should be a linear representation of the same algebra on the product vector space. Imposing such additional structure typically leads to the idea of a
Hopf algebra or a
Lie algebra, as demonstrated below.
Motivation for a Hopf algebra
Consider, for example, two representations
and
. One might try to form a tensor product representation
according to how it acts on the product vector space, so that
:
However, such a map would not be linear, since one would have
:
for ''k'' ∈ ''K''. One can rescue this attempt and restore linearity by imposing additional structure, by defining an algebra homomorphism Δ: ''A'' → ''A'' ⊗ ''A'', and defining the tensor product representation as
:
Such a homomorphism Δ is called a
comultiplication In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams ...
if it satisfies certain axioms. The resulting structure is called a
bialgebra. To be consistent with the definitions of the associative algebra, the coalgebra must be co-associative, and, if the algebra is unital, then the co-algebra must be co-unital as well. A
Hopf algebra is a bialgebra with an additional piece of structure (the so-called antipode), which allows not only to define the tensor product of two representations, but also the Hom module of two representations (again, similarly to how it is done in the representation theory of groups).
Motivation for a Lie algebra
One can try to be more clever in defining a tensor product. Consider, for example,
:
so that the action on the tensor product space is given by
:
.
This map is clearly linear in ''x'', and so it does not have the problem of the earlier definition. However, it fails to preserve multiplication:
:
.
But, in general, this does not equal
:
.
This shows that this definition of a tensor product is too naive; the obvious fix is to define it such that it is antisymmetric, so that the middle two terms cancel. This leads to the concept of a
Lie algebra.
Non-unital algebras
Some authors use the term "associative algebra" to refer to structures which do not necessarily have a multiplicative identity, and hence consider homomorphisms which are not necessarily unital.
One example of a non-unital associative algebra is given by the set of all functions ''f'': R → R whose
limit as ''x'' nears infinity is zero.
Another example is the vector space of continuous periodic functions, together with the
convolution product.
See also
*
Abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
*
Algebraic structure
*
Algebra over a field
*
Sheaf of algebras, a sort of an algebra over a
ringed space
Notes
References
*
*
*
* Nathan Jacobson, Structure of Rings
* James Byrnie Shaw (1907
A Synopsis of Linear Associative Algebra link from
Cornell University
Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to tea ...
Historical Math Monographs.
* Ross Street (1998)
Quantum Groups: an entrée to modern algebra', an overview of index-free notation.
*
{{DEFAULTSORT:Associative Algebra
Algebras
Algebraic geometry